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  • 1.
    Bai, Chengming
    et al.
    Republic of China.
    Fuchs, JürgenKarlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.Huang, Yi-ZhiUSA.Kong, LiangRepublic of China.Runkel, IngoGermany.Schweigert, ChristophGermany.
    Conformal Field Theories and Tensor Categories: Proceedings of a Workshop Held at Beijing International Center for Mathematical Research2014Conference proceedings (editor) (Refereed)
    Abstract [en]

    The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at theBeijing International Center for Mathematical Research, Peking University,  or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.

  • 2.
    Buchberger, Igor
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics. Karlstads Univ, Teoret Fys, S-65188 Karlstad, Sweden..
    Fuchs, Jürgen
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    On the Killing form of Lie Algebras in Symmetric Ribbon Categories2015In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 11, article id 017Article in journal (Refereed)
    Abstract [en]

    As a step towards the structure theory of Lie algebras in symmetric monoidal categories we establish results involving the Killing form. The proper categorical setting for discussing these issues are symmetric ribbon categories.

  • 3. Felder, G.
    et al.
    Fröhlich, J.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    The geometry of WZW branes2000In: Journal of Geom. and Physics 34 (2000)Article in journal (Refereed)
    Abstract [en]

    The structures in target space geometry that correspond to conformally invariant boundary conditions in WZW theories are determined both by studying the scattering of closed string states and by investigating the algebra of open string vertex operators. In the limit of large level, we find branes whose world volume is a regular conjugacy class or, in the case of symmetry breaking boundary conditions, a `twined' version thereof. In particular, in this limit one recovers the commutative algebra of functions over the brane world volume, and open strings connecting different branes disappear. At finite level, the branes get smeared out, yet their approximate localization at (twined) conjugacy classes can be detected unambiguously.

    As a by-product, it is demonstrated how the pentagon identity and tetrahedral symmetry imply that in any rational conformal field theory the structure constants of the algebra of boundary operators coincide with specific entries of fusing matrices

  • 4. Felder, Giovanni
    et al.
    Fröhlich, Jürg
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, Christoph
    Conformal boundary conditions and three-dimensional topological field theory2000In: Physical Review Letters 84 (2000)Article in journal (Refereed)
    Abstract [en]

    We present a general construction of all correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary topologies. The correlators are expressed in terms of Wilson graphs in a certain three-manifold, the connecting manifold. The amplitudes constructed this way can be shown to be modular invariant and to obey the correct factorization rules

  • 5. Felder, Giovanni
    et al.
    Fröhlich, Jürg
    Schweigert, Christoph
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Correlation functions and boundary conditions in RCFT and three-dimensional topology2002In: Compositio mathematica, ISSN 0010-437X, Vol. 131, no 02, p. 189-238Article in journal (Refereed)
    Abstract [en]

    We give a general construction of correlation functions in rational conformal field theory on a possibly non-orientable surface with boundary in terms of 3-dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category

  • 6. Fjelstad, J.
    et al.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    TFT construction of RCFT correlators V: Proof of modular invariance and factorisation2006In: Theory and Appl. of Categories 16 (2006)Article in journal (Refereed)
    Abstract [en]

    The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of [FRSI,FRSII,FRSIV] are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints. We present results both for conformal field theories defined on oriented surfaces and for theories defined on unoriented surfaces. In the latter case, in particular the so-called cross cap constraint is included

  • 7. Fjelstad, J.
    et al.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Topological and conformal field theory as Frobenius algebras2007In: Contemporary Mathematics 431 (2007), 2007Chapter in book (Refereed)
    Abstract [en]

    Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. The construction of a (rational) CFT can be divided into two steps, of which one is complex-analytic and one purely algebraic. We realise the algebraic part of the construction with the help of three-dimensional topological field theory and show that any symmetric special Frobenius algebra in the appropriate braided monoidal category gives rise to a solution. A special class of examples is provided by two-dimensional topological field theories, for which the relevant monoidal category is the category of vector spaces

  • 8. Fjelstad, J.
    et al.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Uniqueness of open/closed rational CFT with given algebra of open states2008In: Adv. in Theor. and Math. Phys. 12 (2008)Article in journal (Refereed)
    Abstract [en]

    We study the sewing constraints for rational two-dimensional conformal field

    theory on oriented surfaces with possibly non-empty boundary. The boundary

    condition is taken to be the same on all segments of the boundary.

    The following uniqueness result is established: For a solution to the sewing

    constraints with nondegenerate closed state vacuum and nondegenerate two-point

    correlators of boundary fields on the disk and of bulk fields on the sphere, up

    to equivalence all correlators are uniquely determined by the one-, two,- and

    three-point correlators on the disk. Thus for any such theory every consistent

    collection of correlators can be obtained by the TFT approach of [hep-th/0204148] and [hep-th/0503194].

    As morphisms of the category of world sheets we include

    not only homeomorphisms, but also sewings; interpreting the correlators

    as a natural transformation then encodes covariance both under homeomorphisms

    and under sewings of world sheets.

  • 9. Fjelstad, Jens
    et al.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Hwang, Stephen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Semikhatov, A.M.
    Tipunin, I.Yu.
    Logarithmic conformal field theories via logarithmic deformations2002In: Nuclear Physics B 633 (2002)Article in journal (Refereed)
  • 10.
    Fjelstad, Jens
    et al.
    Department of Physics, Nanjing University, 22 Hankou Road, Nanjing, 210093, China.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Stigner, Carl
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    RCFT with defects: Factorization and fundamental world sheets2012In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 863, p. 213-259Article in journal (Refereed)
    Abstract [en]

    It is known that for any full rational conformal field theory, the correlation functions that are obtained bythe TFT construction satisfy all locality, modular invariance and factorization conditions, and that there isa small set of fundamental correlators to which all others are related via factorization – provided that theworld sheets considered do not contain any non-trivial defect lines. In this paper we generalize both resultsto oriented world sheets with an arbitrary network of topological defect lines.

  • 11.
    Fjelstad, Jens
    et al.
    Nanjing University, Nanjing, China.
    Fuchs, Jürgen
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Stigner, Carl
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Schweigert, Christoph
    Hamburg University, Germany.
    Partition functions, mapping class groups and Drinfeld doubles2013In: Symmetries and Groups in Contemporary Physics: Proceedings of the XXIX International Colloquium on Group-Theoretical Methods in Physics / [ed] Chengming Bai (Nankai University, China), Jean-Pierre Gazeau (Paris Diderot University, France), Mo-Lin Ge (Nankai University, China), Singapore: World Scientific, 2013, p. 405-410Conference paper (Refereed)
    Abstract [en]

    Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete expressions obtained for the case of Drinfeld doubles of finite groups. The results for doubles are independent of the characteristic of the underlying field, and the general results do not require any assumptions of semisimplicity.

  • 12. Fröhlich, Jürg
    et al.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, Ingo
    Schweigert, Christoph
    Algebras in tensor categories and coset conformal field theories2004In: Fortschritte der Physik, ISSN 0015-8208, E-ISSN 1521-3978, Vol. 52, no 6-7, p. 672-677Article in journal (Refereed)
    Abstract [en]

    The coset construction is the most important tool to construct rational conformal field theories with known chiral data. For some cosets at small level, so-called maverick cosets, the familiar analysis using selection and identification rules breaks down. Intriguingly, this phenomenon is linked to the existence of exceptional modular invariants. Recent progress in CFT, based on studying algebras in tensor categories, allows for a universal construction of the chiral data of coset theories which in particular also applies to maverick cosets

  • 13. Fröhlich, Jürg
    et al.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, Ingo
    Schweigert, Christoph
    Correspondences of ribbon categories2006In: Advances in Mathematics 199 (2006) 192-329Article in journal (Refereed)
    Abstract [en]

    Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.

  • 14.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    On non-semisimple fusion rules and tensor categories2007In: Contemporary Mathematics 442 (2007), 2007Chapter in book (Refereed)
    Abstract [en]

    Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest.

    The fusion rules of C can be block-diagonalized. A conjectural connection between the block-diagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models

  • 15.
    Fuchs, Jürgen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms2006Conference paper (Refereed)
    Abstract [en]

    The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms of Frobenius algebras are introduced; they are inner iff the algebra is symmetric

  • 16.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Abramov, ViktorPaal, EugenStolin, AlexanderTralle, AleksyUrbanski, Pawel
    Algebra, Geometry and Mathematical Physics: the V Baltic-Nordic Workshop "Algebra, Geometry and Mathematical Physics" held at the Be̜dlewo Conference Center on October 12 - 16, 20092011Conference proceedings (editor) (Refereed)
  • 17.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Barmeier, Till
    Runkel, Ingo
    Schweigert, Christoph
    Module categories for permutation modular invariants2010In: International mathematics research notices, ISSN 1073-7928, Vol. 2010, no 16, p. 3067-3100Article in journal (Refereed)
    Abstract [en]

    We show that a braided monoidal category C can be endowed with the structure of a right (and left) module category over C \times C. In fact, there is a family of such module category structures, and they are mutually isomorphic if and only if C allows for a twist. For the case that C is premodular, we compute the internal End of the tensor unit of C, and we show that it is an Azumaya algebra if C is modular. As an application to two-dimensional rational conformal field theory, we show that the module categories describe the permutation modular invariant for models based on the product of two identical chiral algebras. It follows in particular that all permutation modular invariants are physical

  • 18.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Barmeier, Till
    Runkel, Ingo
    Schweigert, Christoph
    On the Rosenberg-Zelinsky sequence in abelian monoidal categories2010In: Journal für die reine und angewandte Mathematik (Crelles Journal), ISSN 0075-4102, no 642, p. 1-36Article in journal (Refereed)
    Abstract [en]

    We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the orbifold construction in conformal field theory.

  • 19.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Caenepeel, StefaanGutt, SimoneSchweigert, ChristophStolin, AlexanderVan Oystaeyen, Freddy
    Noncommutative Structures in Mathematics and Physics2010Conference proceedings (editor) (Refereed)
  • 20.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Fröhlich, J.
    Runkel, I.
    Schweigert, C.
    Duality and defects in rational conformal field theory2007In: Nuclear Physics B 763 (2007)Article in journal (Refereed)
    Abstract [en]

    We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting one-dimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT.

    The case of central charge c=4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states Potts model, is treated as an illustrative example

  • 21.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Fröhlich, J.
    Runkel, I.
    Schweigert, C.
    Kramers-Wannier duality from conformal defects2004In: Physical Review Letters 93 (2004)Article in journal (Refereed)
  • 22.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Fröhlich, Jürg
    Runkel, Ingo
    Schweigert, Christoph
    Defect lines, dualities, and generalised orbifolds2010Conference paper (Refereed)
    Abstract

    Defects are a useful tool in the study of quantum field theories. This is illustrated in the example of two-dimensional conformal field theories. We describe how defect lines and their junction points appear in the description of symmetries and order-disorder dualities, as well as in the orbifold construction and a generalisation thereof that covers exceptional modular invariants

  • 23.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Gaberdiel, M. R.
    Runkel, I.
    Schweigert, C.
    Topological defects for the free boson CFT2007In: Journal of Physics A 40 (2007)Article in journal (Refereed)
    Abstract [en]

    Two different conformal field theories can be joined together along a defect line. We study such defects for the case where the conformal field theories on either side are single free bosons compactified on a circle. We concentrate on topological defects for which the left- and right-moving Virasoro algebras are separately preserved, but not necessarily any additional symmetries. For the case where both radii are rational multiples of the self-dual radius we classify these topological defects. We also show that the isomorphism between two T-dual free boson conformal field theories can be described by the action of a topological defect, and hence that T-duality can be understood as a special type of order-disorder duality

  • 24.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).
    Gannon, Terry
    University of Alberta, Edmonton, Canada.
    Schaumann, Gregor
    Universität Wien, Austria.
    Schweigert, Christoph
    Universität Hamburg, Germany.
    The logarithmic Cardy case: Boundary states and annuli2018In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 930, p. 287-327Article in journal (Refereed)
    Abstract [en]

    We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.

  • 25.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Huiszoon, L.R.
    Schellekens, A.N.
    Schweigert, C.
    Walcher, J.
    Boundaries, crosscaps and simple currents2000In: Physics Letters B 495 (2000)Article in journal (Refereed)
    Abstract [en]

    Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory

  • 26.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Hwang, Stephen
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Semikhatov, A.M.
    Tipunin, I.Y.
    Nonsemisimple fusion algebras and the Verlinde formula2004In: Communications in Mathematical Physics 247 (2004)Article in journal (Refereed)
    Abstract [en]

    We find a nonsemisimple fusion algebra F_p associated with each (1,p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive F_p from modular transformations of characters.

  • 27.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Kaste, P.
    Lerche, W.
    Lütken, A.
    Schweigert, C.
    Walcher, J.
    Boundary fixed points, enhanced gauge symmetry and singular bundles on K32001In: Nuclear Physics B 598 (2001)Article in journal (Refereed)
    Abstract [en]

    We investigate certain fixed points in the boundary conformal field theory representation of type IIA D-branes on Gepner points of K3. They correspond geometrically to degenerate brane configurations, and physically lead to enhanced gauge symmetries on the world-volume. Non-abelian gauge groups arise if the stabilizer group of the fixed points is realized projectively, which is similar to D-branes on orbifolds with discrete torsion. Moreover, the fixed point boundary states can be resolved into several irreducible components. These correspond to bound states at threshold and can be viewed as (non-locally free) sub-sheaves of semi-stable sheaves. Thus, the BCFT fixed points appear to carry two-fold geometrical information: on the one hand they probe the boundary of the instanton moduli space on K3, on the other hand they probe discrete torsion in D-geometry.

  • 28.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Nikolaus, Thomas
    Schweigert, Christoph
    Waldorf, Konrad
    Bundle gerbes and surface holonomy2009Conference paper (Refereed)
  • 29.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Priel, Jan
    Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany..
    Schweigert, Christoph
    Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany..
    Valentino, Alessandro
    Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany..
    On the Brauer Groups of Symmetries of Abelian Dijkgraaf-Witten Theories2015In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 339, no 2, p. 385-405Article in journal (Refereed)
    Abstract [en]

    Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group A, and with vanishing 3-cocycle, is generated by group automorphisms of A, by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of A-bundles, and by partial e-m dualities. We show that transmission functors naturally extracted from extended topological field theories with surface defects give a physical realization of the bijection between invertible bimodule categories of a fusion category and braided auto-equivalences of its Drinfeld center . The latter provides the labels for bulk Wilson lines; it follows that a symmetry is completely characterized by its action on bulk Wilson lines.

  • 30.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    A reason for fusion rules to be even2002In: Journal of Physics A 35 (2002)Article in journal (Refereed)
  • 31.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Boundaries, defects and Frobenius algebras2003Conference paper (Refereed)
    Abstract [en]

    The interpretation of D-branes in terms of open strings has lead to much interest in boundary conditions of two-dimensional conformal field theories (CFTs). These studies have deepened our understanding of CFT and allowed us to develop new computational tools. The construction of CFT correlators based on combining tools from topological field theory and non-commutative algebra in tensor categories, which we summarize in this contribution, allows e.g. to discuss, apart from boundary conditions, also defect lines and disorder fields

  • 32.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Conformal boundary conditions and 3D topological field theory2002Conference paper (Other (popular science, discussion, etc.))
    Abstract [en]

    Topological field theory in three dimensions provides a powerful tool to construct correlation functions and to describe boundary conditions in two-dimensional conformal field theories

  • 33.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Conformal correlation functions, Frobenius algebras and triangulations2002In: Nuclear Physics B 624 (2002)Article in journal (Refereed)
  • 34.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Lie algebras, Fuchsian differential equations and CFT correlation functions2004In: Contemporary Mathematics 343 (2004), American Mathematical Society , 2004Chapter in book (Refereed)
    Abstract [en]

    Affine Kac-Moody algebras give rise to interesting systems of differential equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties of their solutions can be encoded in the structure of a modular tensor category on (a subcategory of) the representation category of the affine Lie

    algebra. We discuss the relation between these solutions and physical correlation functions in two-dimensional conformal field theory. In particular we report on a proof for the existence of the latter on world sheets of arbitrary topology

  • 35.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Open strings and 3d topological field theory2003Conference paper (Other (popular science, discussion, etc.))
  • 36.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Ribbon categories and (unoriented) CFT: Frobenius algebras, automorphisms, reversions2007In: Contemporary Mathematics 431 (2007), 2007Chapter in book (Refereed)
    Abstract [en]

    A Morita class of symmetric special Frobenius algebras A in the modular tensor category of a chiral CFT determines a full CFT on oriented world sheets. For unoriented world sheets, A must in addition possess a reversion, i.e. an isomorphism from A^opp to A squaring to the twist. Any two reversions of an algebra A differ by an element of the group Aut(A) of algebra automorphisms of A. We establish a group homomorphism from Aut(A) to the Picard group of the bimodule category C_AA, with kernel consisting of the inner automorphisms, and we refine Morita equivalence to an equivalence relation between algebras with reversion

  • 37.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    TFT construction of RCFT correlators I: Partition functions2002In: Nuclear Physics B 646 (2002)Article in journal (Refereed)
  • 38.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    TFT construction of RCFT correlators II: Unoriented world sheets2004In: Nuclear Physics B 678 (2004)Article in journal (Refereed)
  • 39.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    TFT construction of RCFT correlators III: Simple currents2004In: Nuclear Physics B 694 (2004)Article in journal (Refereed)
    Abstract [en]

    We use simple currents to construct symmetric special Frobenius algebras in modular tensor categories. We classify such simple current type algebras with the help of abelian group cohomology. We show that they lead to the modular invariant torus partition functions that have been studied by Kreuzer and Schellekens. We also classify boundary conditions in the associated conformal field theories and show that the boundary states are given by the formula proposed in hep-th/0007174. Finally, we investigate conformal defects in these theories.

  • 40.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    TFT construction of RCFT correlators IV: Structure constants and correlation functions2005In: Nuclear Physics B 715 (2005)Article in journal (Refereed)
    Abstract [en]

    We compute the fundamental correlation functions in two-dimensional rational conformal field theory, from which all

    other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap.



    Each of these correlators is presented as the product of a structure constant and the appropriate conformal two- or

    three-point block. The structure constants are expressed as invariants of ribbon graphs in three-manifolds

  • 41.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    The fusion algebra of bimodule categories2008In: Applied Categ. Structures 16 (2008)Article in journal (Refereed)
    Abstract [en]

    We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F.

    As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category

  • 42.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, I.
    Schweigert, C.
    Twining characters and Picard groups in rational conformal field theory2007In: Contemporary Mathematics 442 (2007), 2007Chapter in book (Refereed)
    Abstract [en]

    Picard groups of tensor categories play an important role in rational conformal field theory. The Picard group of the representation category C of a rational vertex algebra can be used to construct examples of (symmetric special) Frobenius algebras in C. Such an algebra A encodes all data needed to ensure the existence of correlators of a local conformal field theory.

    The Picard group of the category of A-bimodules has a physical interpretation, too: it describes internal symmetries of the conformal field theory, and allows one to identify generalized Kramers-Wannier dualities of the theory.

    When applying these general results to concrete models based on affine Lie algebras, a detailed knowledge of certain representations of the modular group is needed. We discuss a conjecture that relates these representations to those furnished by twining characters of affine Lie algebras

  • 43.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Runkel, Ingo
    Germany.
    Schweigert, Christoph
    Germany.
    Twenty-five years of two-dimensional rational conformal field theory2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, no 1Article in journal (Refereed)
  • 44.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Schaumann, Gregor
    Max-Planck-Institut für Mathematik, Bonn.
    Schweigert, Christoph
    Hamburg University.
    A trace for bimodule categories2017In: Applied Categorical Structures, ISSN 0927-2852, E-ISSN 1572-9095, Vol. 25, no 2, p. 227-268Article in journal (Refereed)
    Abstract [en]

    We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued tracefor 1-morphisms in the tricategory of finite tensor categories. It is defined by a universalproperty that is a categorification of Hochschild homology of bimodules over an algebra.We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance.Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Wittentopological field theories.

  • 45.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    Category theory for conformal boundary conditions2003In: Fields Institute Communications 39 (2003), 2003Chapter in book (Refereed)
    Abstract [en]

    We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity.



    We apply our results to the description of boundary conditions in two-dimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex

    operator algebras

  • 46.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    Lie algebra automorphisms in conformal field theory2002In: Contemporary Mathematics 297 (2002), American Mathematical Society , 2002Chapter in book (Refereed)
    Abstract [en]

    The role of automorphisms of infinite-dimensional Lie algebras in conformal field theory is examined. Two main types of applications are discussed; they are related to the enhancement and reduction of symmetry, respectively. The structures one encounters also appear in other areas of physics and mathematics. In particular, they lead to two conjectures on the sub-bundle structure of chiral blocks, and they are instrumental in the study of conformally invariant boundary conditions.

  • 47.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    Solitonic sectors, alpha-induction and symmetry breaking boundaries2000In: Physics Letters B 490 (2000)Article in journal (Refereed)
    Abstract [en]

    We develop a systematic approach to boundary conditions that break bulk symmetries in a general way such that left and right movers are not necessarily connected by an automorphism. In the context of string compactifications, such boundary conditions typically include non-BPS branes. Our formalism is based on two dual fusion rings, one for the bulk and one for the boundary fields. Only in the Cardy case these two structures coincide. In general they are related by a version of alpha-induction. Symmetry breaking boundary conditions correspond to solitonic sectors. In examples, we compute the annulus amplitudes and boundary states

  • 48.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    Symmetry breaking boundaries II. More structures; examples2000In: Nuclear Physics B 568 (2000)Article in journal (Refereed)
    Abstract [en]

    Various structural properties of the space of symmetry breaking boundary conditions that preserve an orbifold subalgebra are established. To each such boundary condition we associate its automorphism type. It is shown that correlation functions in the presence of such boundary conditions are expressible in terms of twisted boundary blocks which obey twisted Ward identities. The subset of boundary conditions that share the same automorphism type is controlled by a classifying algebra, whose structure constants are shown to be traces on spaces of chiral blocks. T-duality on boundary conditions is not a one-to-one map in general. These structures are illustrated in a number of examples. Several applications, including the construction of non-BPS boundary conditions in string theory, are exhibited

  • 49.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    Walcher, J.
    Projections in string theory and boundary states for Gepner models2000In: Nuclear Physics B 588 (2000)Article in journal (Refereed)
    Abstract [en]

    In string theory various projections have to be imposed to ensure supersymmetry. We study the consequences of these projections in the presence of world sheet boundaries. A-type boundary conditions come in several classes; only boundary fields that do not change the class preserve supersymmetry. Our analysis takes in particular properly into account the resolution of fixed points under the projections. Thus e.g. the compositeness of some previously considered boundary states of Gepner models follows from chiral properties of the projections. Our arguments are model independent; in particular, integrality of all annulus coefficients is ensured by model independent arguments

  • 50.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Schweigert, C.
    Waldorf, K.
    Bi-branes: Target space geometry for world sheet topological defects2008In: Journal of Geom. and Physics 58 (2008)Article in journal (Refereed)
    Abstract [en]

    We establish that the relevant geometric data for the target space description of world sheet topological defects are submanifolds - which we call bi-branes - in the product M1 x M2 of the two target spaces involved. Very much like branes, they are equipped with a vector bundle, which in backgrounds with non-trivial B-field is actually a twisted vector bundle. We explain how to define Wess-Zumino terms in the presence of bi-branes and discuss the fusion of bi-branes.

    In the case of WZW theories, symmetry preserving bi-branes are shown to be biconjugacy classes. The algebra of functions on a biconjugacy class is shown to be related, in the limit of large level, to the partition function for defect fields. We finally indicate how the Verlinde algebra arises in the fusion of WZW bi-branes

12 1 - 50 of 75
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